---
id: 5951a53863c8a34f02bf1bdc
title: Problem des dichtesten Punktpaares
challengeType: 1
forumTopicId: 302232
dashedName: closest-pair-problem
---

# --description--

Provide a function to find the closest two points among a set of given points in two dimensions.

Die einfache Lösung ist ein $O(n^2)$-Algorithmus (den wir *Brute-force-Algorithmus* nennen können); der Pseudocode (unter Verwendung von Indizes) könnte einfach sein:

<pre><strong>bruteForceClosestPair</strong> of P(1), P(2), ... P(N)
<strong>if</strong> N &#x3C; 2 <strong>then</strong>
  <strong>return</strong> ∞
<strong>else</strong>
  minDistance ← |P(1) - P(2)|
  minPoints ← { P(1), P(2) }
  <strong>foreach</strong> i ∈ [1, N-1]
    <strong>foreach</strong> j ∈ [i+1, N]
      <strong>if</strong> |P(i) - P(j)| &#x3C; minDistance <strong>then</strong>
        minDistance ← |P(i) - P(j)|
        minPoints ← { P(i), P(j) }
      <strong>endif</strong>
    <strong>endfor</strong>
  <strong>endfor</strong>
  <strong>return</strong> minDistance, minPoints
<strong>endif</strong>
</pre>

Ein besserer Algorithmus basiert auf dem rekursiven Divide-and-Conquer-Ansatz, der $O(n\log n)$ ist, ein Pseudocode sein könnte:

<pre><strong>closestPair</strong> von (xP, yP)
  wobei xP P(1) .. ist P(N) sortiert nach der x-Koordinate, und
  yP ist P(1) .. P(N) sorted by y coordinate (ascending order)
<strong>if</strong> N ≤ 3 <strong>then</strong>
  <strong>return</strong> closest points of xP using brute-force algorithm
<strong>else</strong>
  xL ← points of xP from 1 to ⌈N/2⌉
  xR ← points of xP from ⌈N/2⌉+1 to N
  xm ← xP(⌈N/2⌉)<sub>x</sub>
  yL ← { p ∈ yP : p<sub>x</sub> ≤ xm }
  yR ← { p ∈ yP : p<sub>x</sub> > xm }
  (dL, pairL) ← closestPair of (xL, yL)
  (dR, pairR) ← closestPair of (xR, yR)
  (dmin, pairMin) ← (dR, pairR)
  <strong>if</strong> dL &#x3C; dR <strong>then</strong>
    (dmin, pairMin) ← (dL, pairL)
  <strong>endif</strong>
  yS ← { p ∈ yP : |xm - p<sub>x</sub>| &#x3C; dmin }
  nS ← number of points in yS
  (closest, closestPair) ← (dmin, pairMin)
  <strong>for</strong> i <strong>from</strong> 1 <strong>to</strong> nS - 1
    k ← i + 1
    <strong>while</strong> k ≤ nS <strong>and</strong> yS(k)<sub>y</sub> - yS(i)<sub>y</sub> &#x3C; dmin
      <strong>if</strong> |yS(k) - yS(i)| &#x3C; closest <strong>then</strong>
        (closest, closestPair) ← (|yS(k) - yS(i)|, {yS(k), yS(i)})
      <strong>endif</strong>
      k ← k + 1
    <strong>endwhile</strong>
  <strong>endfor</strong>
  <strong>return</strong> closest, closestPair
<strong>endif</strong>
</pre>

Für die Eingabe wird erwartet, dass das Argument ein Array von `Point`-Objekten mit `x` und `y` Mitglieder auf Zahlen gesetzt. Gibt ein Objekt zurück, das die Schlüssel:Wert-Paare für `distance` und `pair` (das Paar der beiden nächstgelegenen Punkte) enthält.

Zum Beispiel `getClosestPair` mit dem Eingabefeld `points`:

```js
const points = [
  new Point(1, 2),
  new Point(3, 3),
  new Point(2, 2)
];
```

Würde zurückgeben:

```js
{
  distance: 1,
  pair: [
    {
      x: 1,
      y: 2
    },
    {
      x: 2,
      y: 2
    }
  ]
}
```

**Note:** sortiere die `pair`-Anordnung nach ihren `x`-Werten in aufsteigender Reihenfolge.


# --hints--

`getClosestPair` sollte eine Funktion sein.

```js
assert(typeof getClosestPair === 'function');
```

`getClosestPair(points1).distance` sollte `0.0894096443343775` sein.

```js
assert.equal(getClosestPair(points1).distance, answer1.distance);
```

`getClosestPair(points1).pair` sollte `[ { x: 7.46489, y: 4.6268 }, { x: 7.46911, y: 4.71611 } ]` sein.

```js
assert.deepEqual(
  JSON.parse(JSON.stringify(getClosestPair(points1))).pair,
  answer1.pair
);
```

`getClosestPair(points2).distance` sollte `65.06919393998976` sein.

```js
assert.equal(getClosestPair(points2).distance, answer2.distance);
```

`getClosestPair(points2).pair` sollte `[ { x: 37134, y: 1963 }, { x: 37181, y: 2008 } ]` sein.

```js
assert.deepEqual(
  JSON.parse(JSON.stringify(getClosestPair(points2))).pair,
  answer2.pair
);
```

`getClosestPair(points3).distance` sollte `6754.625082119658` sein.

```js
assert.equal(getClosestPair(points3).distance, answer3.distance);
```

`getClosestPair(points3).pair` sollte `[ { x: 46817, y: 64975 }, { x: 48953, y: 58567 } ]` sein.

```js
assert.deepEqual(
  JSON.parse(JSON.stringify(getClosestPair(points3))).pair,
  answer3.pair
);
```

# --seed--

## --after-user-code--

```js
const points1 = [
    new Point(0.748501, 4.09624),
    new Point(3.00302, 5.26164),
    new Point(3.61878,  9.52232),
    new Point(7.46911,  4.71611),
    new Point(5.7819,   2.69367),
    new Point(2.34709,  8.74782),
    new Point(2.87169,  5.97774),
    new Point(6.33101,  0.463131),
    new Point(7.46489,  4.6268),
    new Point(1.45428,  0.087596)
];

const answer1 = {
  distance: 0.0894096443343775,
  pair: [
    {
      x: 7.46489,
      y: 4.6268
    },
    {
      x: 7.46911,
      y: 4.71611
    }
  ]
};

const points2 = [
  new Point(37100, 13118),
  new Point(37134, 1963),
  new Point(37181, 2008),
  new Point(37276, 21611),
  new Point(37307, 9320)
];

const answer2 = {
  distance: 65.06919393998976,
  pair: [
    {
      x: 37134,
      y: 1963
    },
    {
      x: 37181,
      y: 2008
    }
  ]
};

const points3 = [
  new Point(16910, 54699),
  new Point(14773, 61107),
  new Point(95547, 45344),
  new Point(95951, 17573),
  new Point(5824, 41072),
  new Point(8769, 52562),
  new Point(21182, 41881),
  new Point(53226, 45749),
  new Point(68180, 887),
  new Point(29322, 44017),
  new Point(46817, 64975),
  new Point(10501, 483),
  new Point(57094, 60703),
  new Point(23318, 35472),
  new Point(72452, 88070),
  new Point(67775, 28659),
  new Point(19450, 20518),
  new Point(17314, 26927),
  new Point(98088, 11164),
  new Point(25050, 56835),
  new Point(8364, 6892),
  new Point(37868, 18382),
  new Point(23723, 7701),
  new Point(55767, 11569),
  new Point(70721, 66707),
  new Point(31863, 9837),
  new Point(49358, 30795),
  new Point(13041, 39744),
  new Point(59635, 26523),
  new Point(25859, 1292),
  new Point(1551, 53890),
  new Point(70316, 94479),
  new Point(48549, 86338),
  new Point(46413, 92747),
  new Point(27186, 50426),
  new Point(27591, 22655),
  new Point(10905, 46153),
  new Point(40408, 84202),
  new Point(52821, 73520),
  new Point(84865, 77388),
  new Point(99819, 32527),
  new Point(34404, 75657),
  new Point(78457, 96615),
  new Point(42140, 5564),
  new Point(62175, 92342),
  new Point(54958, 67112),
  new Point(4092, 19709),
  new Point(99415, 60298),
  new Point(51090, 52158),
  new Point(48953, 58567)
];

const answer3 = {
  distance: 6754.625082119658,
  pair: [
    {
      x: 46817,
      y: 64975
    },
    {
      x: 48953,
      y: 58567
    }
  ]
}
```

## --seed-contents--

```js
const Point = function(x, y) {
  this.x = x;
  this.y = y;
};
Point.prototype.getX = function() {
  return this.x;
};
Point.prototype.getY = function() {
  return this.y;
};

function getClosestPair(pointsArr) {

  return true;
}
```

# --solutions--

```js
const Point = function(x, y) {
  this.x = x;
  this.y = y;
};
Point.prototype.getX = function() {
  return this.x;
};
Point.prototype.getY = function() {
  return this.y;
};

const mergeSort = function mergeSort(points, comp) {
    if(points.length < 2) return points;

    var n = points.length,
        i = 0,
        j = 0,
        leftN = Math.floor(n / 2),
        rightN = leftN;

    var leftPart = mergeSort( points.slice(0, leftN), comp),
        rightPart = mergeSort( points.slice(rightN), comp );

    var sortedPart = [];

    while((i < leftPart.length) && (j < rightPart.length)) {
        if(comp(leftPart[i], rightPart[j]) < 0) {
            sortedPart.push(leftPart[i]);
            i += 1;
        }
        else {
            sortedPart.push(rightPart[j]);
            j += 1;
        }
    }
    while(i < leftPart.length) {
        sortedPart.push(leftPart[i]);
        i += 1;
    }
    while(j < rightPart.length) {
        sortedPart.push(rightPart[j]);
        j += 1;
    }
    return sortedPart;
};

const closestPair = function _closestPair(Px, Py) {
    if(Px.length < 2) return { distance: Infinity, pair: [ new Point(0, 0), new Point(0, 0) ] };
    if(Px.length < 3) {
        //find euclid distance
        var d = Math.sqrt( Math.pow(Math.abs(Px[1].x - Px[0].x), 2) + Math.pow(Math.abs(Px[1].y - Px[0].y), 2) );
        return {
            distance: d,
            pair: [ Px[0], Px[1] ]
        };
    }

    var n = Px.length,
        leftN = Math.floor(n / 2),
        rightN = leftN;

    var Xl = Px.slice(0, leftN),
        Xr = Px.slice(rightN),
        Xm = Xl[leftN - 1],
        Yl = [],
        Yr = [];
    //separate Py
    for(var i = 0; i < Py.length; i += 1) {
        if(Py[i].x <= Xm.x)
            Yl.push(Py[i]);
        else
            Yr.push(Py[i]);
    }

    var dLeft = _closestPair(Xl, Yl),
        dRight = _closestPair(Xr, Yr);

    var minDelta = dLeft.distance,
        closestPair = dLeft.pair;
    if(dLeft.distance > dRight.distance) {
        minDelta = dRight.distance;
        closestPair = dRight.pair;
    }

    //filter points around Xm within delta (minDelta)
    var closeY = [];
    for(i = 0; i < Py.length; i += 1) {
        if(Math.abs(Py[i].x - Xm.x) < minDelta) closeY.push(Py[i]);
    }
    //find min within delta. 8 steps max
    for(i = 0; i < closeY.length; i += 1) {
        for(var j = i + 1; j < Math.min( (i + 8), closeY.length ); j += 1) {
            var d = Math.sqrt( Math.pow(Math.abs(closeY[j].x - closeY[i].x), 2) + Math.pow(Math.abs(closeY[j].y - closeY[i].y), 2) );
            if(d < minDelta) {
                minDelta = d;
                closestPair = [ closeY[i], closeY[j] ]
            }
        }
    }

    return {
        distance: minDelta,
        pair: closestPair.sort((pointA, pointB) => pointA.x - pointB.x)
    };
};

function getClosestPair(points) {
  const sortX = function(a, b) { return (a.x < b.x) ? -1 : ((a.x > b.x) ? 1 : 0); }
  const sortY = function(a, b) { return (a.y < b.y) ? -1 : ((a.y > b.y) ? 1 : 0); }

  const Px = mergeSort(points, sortX);
  const Py = mergeSort(points, sortY);

  return closestPair(Px, Py);
}
```
